BGE 2.79: How do I change an action in-game with the push of a button/object
The always sensor (with the controller) activates/deactivates the action actuator. It does not change the configuration of the actuator.

As it is now it reads like this:

Always (due to True Level Triggering) -> activate the actuator

This means at each single frame you tell the actuator to play the action. 

You wrote you want to play another action. To do so you need another action actuator that does exactly that. 

Now you need a way to determine when to play what action. There are mutliple ways to achieve that. Here is one.

You define two modes: playing A and playing B
You start with mod playing A.

Add a property call it "mode" make it String and Initialize it with "playing A".

Now change your always sensor to a property sensor listening on property "mode" to be equal to "playing A". 

Right now it should work as before. You start the game and the object plays A.

(Hint: switch the action actuator to loop stop mode and you can disable the true level triggering at the property sensor)

Now add another Property sensor that checks "mode" to be equal to "playing B". Connect to a controller, connect to another action actuator defining the other action.

So this is what you wanted to get. A way to play another action. 

Now you need to control what action to play. I assume you send a message "play B" at the moment you want to see the second action.

Add a message sensor that listens to "play B", connect to a controller, connect to a property actuator that changes "mode" to "playing B".

With that you can change the animation in-game. Just send "play B" and the object will stop playing A (dependent on the actuator mode) and starts playing B.

To switch back you need to add a similar logic listening to another message subject (or the same if you want to toggle).